Question

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through $O$ (the centre of mass) and $O^{\prime }$ (corner point) is :

• A. $\frac{2}{3}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $\frac{1}{4}$

The moment of inertia of a rectangular sheet, about an axis passing through $O\left(\text{centre of mass}\right)$ and perpendicular to its plane, is,

$I_O=\frac{M}{12}(a^2+b^2)=\frac{M}{12}\left[\right(80{)}^2+\left(60{)}^2\right]$


According to parallel axis's theorem, moment of inertia about $O^{\prime }$ is,

$I_{0^{\prime }}=I_O+M(50{)}^2$

$\therefore \frac{I_0}{I_{0^{\prime }}}=\frac{\frac{M}{12}\left[\right(80{)}^2+\left(60{)}^2\right]}{\frac{M}{12}\left[\right(80{)}^2+\left(60{)}^2\right]+M(50{)}^2}=\frac{1}{4}$

Hence, $\left(B\right)$ is the correct answer.